Moduli Spaces of Semistable Sheaves on Singular Genus One Curves

TL;DR

This paper explores the moduli spaces of semistable sheaves on singular genus one curves, establishing derived equivalences that preserve stability and identifying these moduli spaces with symmetric products or other known geometric objects.

Contribution

It introduces new derived equivalences that preserve stability and provides explicit descriptions of moduli spaces for various types of genus one curves, including singular and reducible cases.

Findings

Moduli spaces for rank zero are symmetric powers of the curve.

Finite non-isomorphic moduli spaces for fixed positive rank.

Connected components of moduli spaces for vector bundles relate to symmetric products of rational curves.

Abstract

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of non-isomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle $E_N$ of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves $\mathcal{O}(-1)$ supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank $r$ is isomorphic to the $r$-th symmetric product of the rational curve with one node.